Topological types and analytic invariants of complex surface singularities

• Project/Area Number: 17K05216
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Yamagata University
• Principal Investigator: Okuma Tomohiro 山形大学, 理学部, 教授 (00300533)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥3,770,000 (Direct Cost: ¥2,900,000、Indirect Cost: ¥870,000); Fiscal Year 2020: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2018: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000); Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
• Keywords: 2次元特異点 / 幾何種数 / 2次元正規特異点 / 複素2次元特異点 / 正規節減数 / Brieskorn 完全交叉特異点 / 特異点

Outline of Final Research Achievements

The aim of this research is to study fundamental analytic invariants and the structure of normal surface singularities. For Brieskorn complete intersection singularities, we obtained a formula of the normal reduction number of maximal ideals and classified the elliptic singularities. We gave examples of certain distinguished structures of singularities which are homeomorphic to a Brieskorn complete intersection singularity. For cone singularities, we obtained a simple formula of the normal reduction number. We also introduced elliptic ideals and strongly elliptic ideals and obtained their basic properties.

Academic Significance and Societal Importance of the Research Achievements

代数多様体や複素解析空間には特異点が存在する．特異点の性質を捉えることで，それらの深い理解につながることがある．本研究は2次元特異点を対象に，基本的な解析的不変量や特異点の構造について，より詳しい結果を得るとともに，新たな研究課題を見出している．これらの成果はこれからの研究の進展に寄与するものと思われる．

Research Products

• [Int'l Joint Research] University of Genoa(イタリア)
• [Int'l Joint Research] Alfred Renyi Institute of Mathematics(ハンガリー)
• [Int'l Joint Research] レニイ数学研究所(ハンガリー)
• [Journal Article] Normal Hilbert coefficients and elliptic ideals in normal two-dimensional singularities (2022)
  • Author(s): Okuma, Tomohiro and Rossi, Maria Evelina and Watanabe, Kei-ichi and Yoshida, Ken-ichi
  • Journal Title: Nagoya Mathematical Journal Volume: - Pages: 1-22
  • DOI: 10.1017/nmj.2022.5
  • Peer Reviewed / Open Access / Int'l Joint Research
• [Journal Article] Normal reduction numbers of normal surface singularities (2021)
  • Author(s): Okuma, Tomohiro
  • Journal Title: Singularities and their interaction with geometry and low dimensional topology. In honor of Andras Nemethi on the occasion of his 60th birthday. Volume: - Pages: 159-179
  • DOI: 10.1007/978-3-030-61958-9_8
  • ISBN: 9783030619572, 9783030619589
  • Peer Reviewed / Open Access
• [Journal Article] The normal reduction number of two-dimensional cone-like singularities (2021)
  • Author(s): Okuma, Tomohiro and Watanabe, Kei-ichi and Yoshida, Ken-ichi
  • Journal Title: Proceedings of the American Mathematical Society Volume: 149 (no. 11) Issue: 11 Pages: 4569-4581
  • DOI: 10.1090/proc/15565
  • Peer Reviewed / Open Access
• [Journal Article] Weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersections (2021)
  • Author(s): Okuma, Tomohiro
  • Journal Title: Journal of Singularities Volume: 23
  • DOI: 10.5427/jsing.2021.23j
  • Peer Reviewed / Open Access
• [Journal Article] The normal reduction number of two-dimensional cone-like singularities (2021)
  • Author(s): Tomohiro Okuma, Kei-ichi Watanabe, and Ken-ichi Yoshida
  • Journal Title: Proceedings of the AMS Volume: -
  • Peer Reviewed / Open Access
• [Journal Article] Normal Reduction Numbers for Normal Surface Singularities with Application to Elliptic Singularities of Brieskorn Type (2019)
  • Author(s): Okuma Tomohiro、Watanabe Kei-ichi、Yoshida Ken-ichi
  • Journal Title: Acta Mathematica Vietnamica Volume: 44 Issue: 1 Pages: 87-100
  • DOI: 10.1007/s40306-018-00311-4
  • Peer Reviewed / Open Access
• [Journal Article] Cohomology of ideals in elliptic surface singularities (2018)
  • Author(s): T. Okuma
  • Journal Title: Illinois J. Math. Volume: 61
  • Peer Reviewed / Open Access
• [Journal Article] A characterization of two-dimensional rational singularities via core of ideals (2018)
  • Author(s): Okuma Tomohiro、Watanabe Kei-ichi、Yoshida Ken-ichi
  • Journal Title: J. Algebra Volume: 499 Pages: 450-468
  • DOI: 10.1016/j.jalgebra.2017.11.053
  • Peer Reviewed / Open Access
• [Journal Article] Analytic singularities supported by a specific integral homology sphere link (2017)
  • Author(s): A. Nemethi and T. Okuma
  • Journal Title: Methods Appl. Anal. Volume: 24, No. 2 Issue: 2 Pages: 303-320
  • DOI: 10.4310/maa.2017.v24.n2.a7
  • Peer Reviewed / Int'l Joint Research
• [Presentation] 複素２次元特異点の基本的な解析的不変量について (2022)
  • Author(s): 奥間智弘
  • Organizer: 日本数学会東北支部会
  • Invited
• [Presentation] Normal reduction numbers of normal surface singularities (2019)
  • Author(s): Tomohiro Okuma
  • Organizer: Geometry and Topology of Singularities--- In honor of Andras Nemethi's 60th birthday
  • Int'l Joint Research / Invited
• [Presentation] Normal reduction numbers and cohomology of normal surface singularities (2019)
  • Author(s): Tomohiro Okuma
  • Organizer: Algebraic surfaces and related topics---In honor of Kazuhiro Konno's 60th birthday
  • Int'l Joint Research / Invited
• [Presentation] Weighted homogeneous surface singularities homeomorphic to Brieskorn complete intersections (2019)
  • Author(s): 奥間智弘
  • Organizer: 日本数学会
• [Presentation] Cohomology of ideals in normal surface singularities (2017)
  • Author(s): T. Okuma
  • Organizer: International Conference on Singularity Theory
  • Int'l Joint Research / Invited
• [Presentation] 位相型が等しくタイプが異なる複素2次元特異点について (2017)
  • Author(s): 奥間智弘
  • Organizer: 第60回函数論シンポジウム
  • Invited
• [Presentation] Cohomology of ideal sheaves on resolutions of surface singularities (2017)
  • Author(s): 奥間智弘
  • Organizer: 2017年度多変数関数論冬セミナー
  • Invited